The 10V 6800uF caps negative is connected to the negative of a 9V battery. The positive is connected to a 2 pin IC, labeled S14, which in turn is connected to an 8 pin surface mount IC labeled ND5 8410A.
Im not sure exactly what each pins number is (on the ND5), but I will guess. And I cant say where some of the traces go since I cant see the middle layer of the PCB. But what I do know is pins 1-3 share a trace, pin 4 has its own trace, the S14 IC is soldered to pin 5 and 6, and it looks like pin 7 and 8 share a trace that goes to one of the pins of the solenoid. However, I cant tell if pins 5/6 are on the same trace as 7/8.
Also, I brought the solenoid to work today and tried various lower voltages. The solenoid operated at very low voltages (I think it worked at like 2 or 3V), but with each decrease in voltage came a noticeable decrease in power. However, it did still seem quite powerful at 5V, but I dont feel that my perception of its power at 5V is proof that this solenoid actually operated at that voltage originally.